Question: Consider the following polynomial function. f(x) = x^4 + x^3 - 7x^2 - 5x + 10 Step 3 of 4: Find…

Consider the following polynomial function.

f(x) = x^4 + x^3 - 7x^2 - 5x + 10

Step 3 of 4: Find the x-intercept(s) at which f crosses the axis. Express the intercept(s) as ordered pair(s).

Select the number of x-intercept(s) at which f crosses the axis.

Selecting an option will display any text boxes needed to complete your answer.

none 1 2 3 4

Solution

Let’s solve the problem step by step. First, we need to find the x-intercepts of the polynomial function: \[ f(x) = x^4 + x^3 - 7x^2 - 5x + 10 \] To find the x-intercepts, we set \( f(x) = 0 \): \[ x^4 + x^3 - 7x^2 - 5x + 10 = 0 \] We can use the Rational Root Theorem to find possible rational roots. The possible rational roots are the factors of the constant term (10) divided by the factors of the leading coefficient (1): \[ \text{Possible roots}: \pm1, \pm2, \pm5, \pm10 \] Let’s test these possible roots: 1. Testing \( x = 1 \): \[ f(1) = 1^4 + 1^3 - 7(1)^2 - 5(1) + 10 = 1 + 1 - 7 - 5 + 10 = 0 \] Since \( f(1) = 0 \), \( x = 1 \) is a root. Next, we perform synthetic division to factor out \( (x - 1) \) from the polynomial: \[ \begin{array}{r|rrrr} 1 & 1 & 1 & -7 & -5 & 10 \\ & & 1 & 2 & -5 & -10 \\ \hline & 1 & 2 & -5 & -10 & 0 \\ \end{array} \] The quotient is: \[ x^3 + 2x^2 - 5x - 10 \] Now, we factor the cubic polynomial \( x^3 + 2x^2 - 5x - 10 \). We test the remaining possible rational roots: 2. Testing \( x = -2 \): \[ f(-2) = (-2)^3 + 2(-2)^2 - 5(-2) - 10 = -8 + 8 + 10 - 10 = 0 \] Since \( f(-2) = 0 \), \( x = -2 \) is a root. We perform synthetic division again to factor out \( (x + 2) \): \[ \begin{array}{r|rrrr} -2 & 1 & 2 & -5 & -10 \\ & & -2 & 0 & 10 \\ \hline & 1 & 0 & -5 & 0 \\ \end{array} \] The quotient is: \[ x^2 - 5 \] Setting \( x^2 - 5 = 0 \), we solve for \( x \): \[ x^2 - 5 = 0 \] \[ x^2 = 5 \] \[ x = \sqrt{5}, \quad x = -\sqrt{5} \] So, the x-intercepts are: \[ (1, 0), \quad (-2, 0), \quad (\sqrt{5}, 0), \quad (-\sqrt{5}, 0) \] Therefore, the polynomial function \( f(x) \) has 4 x-intercepts where it crosses the axis.